The \(2+1\) Kepler problem and its quantization (Q2752489)

The system of two pointlike particles coupled to three-dimensional Einstein gravity is studied. Before the gravitational interaction is switched on, a system of two uncoupled relativistic point particles in flat, three-dimensional Minkowski space is considered. The relativistic definition of the center-of-mass frame is given, which is actually a center of energy frame. After imposing the appropriate restriction on the phase space the authors go through the usual canonical programme. Starting from the classical Hamiltonian framework they perform a phase space reduction, derive and solve the classical equations of motion, and finally they quantize it deriving the energy eigenstates and the spectra of certain interesting operators. At the classical level, the free-particle system also provides a nice toy model for the Hamilton formulation of the general relativity within the ADM framework. At the quantum level, they consider two alternative quantization methods, the Schrödinger method applied to the reduced classical phase space, where all the gauge symmetries are removed, and the Dirac method applied to an extended phase space, where the dynamics of the system is defined by a generalized mass-shell constraint. It is possible to set up a well defined operator representation, and to quantize and solve the constraint equation, which is a generalized Klein-Gordon equation. The authors solve this constraint equation and finally express the energy eigenstates of the Kepler system explicitly as wavefunctions on a suitable defined configuration space, i.e. the wave function has the usual physical interpretation as a probability amplitude for the particle in the space. A quantization of geometry, which restricts the possible asymptotic geometries of the universe is also obtained.